Remote Access Bulletin of the American Mathematical Society

Bulletin of the American Mathematical Society

ISSN 1088-9485(online) ISSN 0273-0979(print)

 
 

 

Nonlocal invariants in index theory


Author: Steven Rosenberg
Journal: Bull. Amer. Math. Soc. 34 (1997), 423-433
MSC (1991): Primary 58G25; Secondary 58G10, 58G25, 58G26
DOI: https://doi.org/10.1090/S0273-0979-97-00731-3
MathSciNet review: 1458426
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In its original form, the Atiyah-Singer Index Theorem equates two global quantities of a closed manifold, one analytic (the index of an elliptic operator) and one topological (a characteristic number). Because it relates invariants from different branches of mathematics, the Index Theorem has many applications and extensions to differential geometry, K-theory, mathematical physics, and other fields. This report focuses on advances in geometric aspects of index theory. For operators naturally associated to a Riemannian metric on a closed manifold, the topological side of the Index Theorem can often be expressed as the integral of local (i.e. pointwise) curvature expression. We will first discuss these local refinements in §1, which arise naturally in heat equation proofs of the Index Theorem. In §§2,3, we discuss further developments in index theory which lead to spectral invariants, the eta invariant and the determinant of an elliptic operator, that are definitely nonlocal. Finally, in §4 we point out some recent connections among these nonlocal invariants and classical index theory.


References [Enhancements On Off] (What's this?)

  • 1. M. F. Atiyah, R. Bott, and V. K. Patodi, On the heat equation and the index theorem, Inventiones Math. 19 (1973), 279-330. MR 58:31287
  • 2. M. F. Atiyah, H. Donnelly, and I. M. Singer, Eta invariants, signature defects of cusps and values of L-functions, Ann. Math. 118 (1983), 131-177. MR 86g:58134a
  • 3. M. F. Atiyah, V. K. Patodi, and I. M. Singer, Spectral asymmetry and Riemannian geometry. I, Math. Proc. Camb. Phil. Soc. 77 (1975), 43-69. MR 53:1655a
  • 4. -, Spectral asymmetry and Riemannian geometry. III, Math. Proc. Camb. Phil. Soc. 79 (1976), 71-99. MR 53:1655c
  • 5. M. F. Atiyah and I. M. Singer, The index of elliptic operators on compact manifolds, Bull. Amer. Math. Soc. 69 (1963), 422-433. MR 28:626
  • 6. -, The index of elliptic operators. III, Ann. Math. 87 (1968), 546-604. MR 38:5245
  • 7. -, Dirac operators coupled to vector potentials, Proc. Nat. Acad. Sci., USA 81 (1984), 2597-2600. MR 86g:58127
  • 8. N. Berline, E. Getzler, and M. Vergne, Heat Kernels and Dirac Operators, Grundlehren der mathematischen Wissenschaften, vol. 298, Springer-Verlag, Berlin, 1992. MR 94e:58130
  • 9. J.-M. Bismut, The Atiyah-Singer index theorem for families of Dirac operators: two heat equation proofs, Inventiones Math. 83 (1986), 91-151. MR 87g:58117
  • 10. J.-M. Bismut and J. Cheeger, Families index for manifolds with boundary, superconnections, and cones. I. Families of manifolds with boundary and Dirac operators, J. Funct. Anal. 89 (1990), 313-363. MR 91e:58180
  • 11. -, Families index for manifolds with boundary, superconnections, and cones. II. The Chern character, J. Funct. Anal. 90 (1990), 306-354. MR 91e:58181
  • 12. -, Transgressed Euler classes of SL(2n,Z) vector bundles, adiabatic limits of eta invariants and special values of L-functions, Ann. scient. Éc. Norm. Sup. 25 (1992), 335-391. MR 94e:57042
  • 13. J.-M. Bismut and D. Freed, The analysis of elliptic families I: Metrics and connections on determinant line bundles, Commun. Math. Phys. 106 (1986), 159-176. MR 88h:58110a
  • 14. -, The analysis of elliptic families II: Dirac operators, eta invariants, and the holonomy theorem of Witten, Commun. Math. Phys. 107 (1986), 103-163. MR 88h:58110b
  • 15. J.-M. Bismut, H. Gillet, and C. Soulé, Analytic torsion and holomorphic determinant bundles. I-III, Commun. Math. Phys. 115 (1988), 49-78, 79-126, 301-351. MR 89g:58192a/b/c
  • 16. J.-M. Bismut and J. Lott, Flat vector bundles, direct images, and higher real analytic torsion, J. Amer. Math. Soc. 8 (1995), 291-363. MR 96g:58202
  • 17. J.-M. Bismut and W. Zhang, An extension of a theorem of Cheeger and Müller, Astérisque 205 (1992), 3-235. MR 93j:58138
  • 18. T. Branson and B. Ørsted, Conformal geometry and global invariants, Diff. Geom. Appl. 1 (1991), 279-308. MR 94k:58154
  • 19. D. Burghelea, L. Friedlander, and T. Kappeler, Torsions for manifolds with boundary and gluing formulas, preprint.
  • 20. A. Carey and V. Mathai, ${L}^2$-acyclicity and ${L}^2$-torsion invariants, Contemporary Mathematics 105 (1990), American Mathematical Society, Providence, RI, pps. 91-118. MR 91e:58187
  • 21. J. Cheeger, Analytic torsion and the heat equation, Ann. Math. 109 (1979), 259-322. MR 80j:58065a
  • 22. X. Dai and D. Freed, $\eta$-invariants and determinant lines, J. Math. Phys. 35 (1994), 5155-5194. MR 96a:58204
  • 23. W. Dwyer, M. Weiss, and B. Williams, A parametrized index theorem for the algebraic K-theory Euler class, preprint, http:/www.math.uiuc.edu/K-theory/0086/(1995).
  • 24. E. Getzler, A short proof of the local Atiyah-Singer Index Theorem, Topology 25 (1986), 111-117. MR 87h:58207
  • 25. P. B. Gilkey, Curvature and the eigenvalues of the Laplacian for elliptic complexes, Adv. in Math. 10 (1973), 344-381. MR 48:3081
  • 26. -, Invariance Theory, the Heat Equation, and the Atiyah-Singer Index Theorem, Publish or Perish, Wilmington, DE, 1984. MR 86j:58144
  • 27. J. Heitsch and C. Lazarov, Riemann-Roch-Grothendieck and torsion for foliations, preprint.
  • 28. K. Igusa, Parametrized Morse theory and its applications, Proc. Int. Cong. Math., Kyoto 1990, Mathematical Society of Japan, Tokyo (1991), 643-651. MR 93c:57022
  • 29. F. Kamber and P. Tondeur, Characteristic invariants of foliated bundles, Manuscripta Math. 11 (1974), 51-89. MR 48:12556
  • 30. J. Klein, Higher Franz-Reidemeister torsion: low-dimensional applications, Contemporary Mathematics 150 (1993), American Mathematical Society, Providence, RI, pps. 195-204. MR 94g:19004
  • 31. J. Lott, Diffeomorphisms, analytic torsion and noncommutative geometry, dg-ga/9607006.
  • 32. -, Heat kernels on covering spaces and topological invariants, J. Differential Geometry 35 (1992), 471-510. MR 93b:58140
  • 33. R. B. Melrose, The Atiyah-Patodi-Singer Index Theorem, A. K. Peters, Wellesley, MA, 1993. MR 96g:58180
  • 34. J. Milnor, Whitehead torsion, Bull. Amer. Math. Soc. 72 (1966), 358-426. MR 33:4922
  • 35. W. Müller, Analytic torsion and R-torsion of Riemannian manifolds, Adv. in Math. 28 (1978), 233-305. MR 80j:58065b
  • 36. -, L${}^2$ index theory, eta invariants and values of L-functions, Contemporary Mathematics 105 (1990), American Mathematical Society, Providence, RI, pps. 141-190. MR 91g:58274
  • 37. B. Osgood, R. Phillips, and P. Sarnak, Extremals of determinants of Laplacians, J. Funct. Anal. 80 (1988), 148-211. MR 90d:58159
  • 38. R. Palais (ed.), Seminar on the Atiyah-Singer Index Theorem, Annals of Mathematics Study, vol. 57, Princeton University Press, Princeton, 1965. MR 33:6649
  • 39. T. Parker and S. Rosenberg, Invariants of conformal Laplacians, J. Differential Geometry 25 (1987), 199-222. MR 89e:58118
  • 40. V. K. Patodi, An analytic proof of the Riemann-Roch-Hirzebruch theorem for Kähler manifolds, J. Differential Geometry 5 (1971), 251-283. MR 44:7502
  • 41. D. Quillen, Determinants of Cauchy-Riemann operators on Riemann surfaces, Funk. Anal. i Prilozhen 19 (1985), 37-41. MR 86g:32035
  • 42. -, Superconnections and the Chern character, Topology 24 (1985), 89-95. MR 86m:58010
  • 43. D. B. Ray and I. M. Singer, R-torsion and the Laplacian on Riemannian manifolds, Adv. in Math. 7 (1971), 145-210. MR 45:4447
  • 44. J. Roe, Elliptic Operators, Topology, and Asymptotic Methods, Pitman Research Notes in Mathematics, vol. 179, Longman Scientific and Technical, Burnt Mill, UK, 1988. MR 89j:58126
  • 45. -, Coarse cohomology and index theory on complete Riemannian manifolds, Memoirs of the Amer. Math. Soc., 497, 1993. MR 94a:58193
  • 46. S. Rosenberg, The determinant of a conformally covariant operator, J. London Math. Soc. 36 (1987), 553-568. MR 89h:58205
  • 47. -, The Laplacian on a Riemannian Manifold, Cambridge U. Press, Cambridge, UK, 1997.
  • 48. R. Seeley, Complex powers of an elliptic operator, Proc. Symp. Pure Math., vol. 10, Amer. Math. Soc., Providence, RI, 1967, pp. 288-307. MR 38:6220
  • 49. I. M. Singer, The $\eta$-invariant and the index, Mathematical Aspects of String Theory (S. T. Yau, ed.), World Scientific Press, Singapore, 1987, pp. 239-258. CMP 20:04
  • 50. S. Vishik, Generalized Ray-Singer conjecture. I. A manifold with a smooth boundary, Commun. Math. Phys. 167 (1995), 1-102. MR 96f:58184
  • 51. E. Witten, Global gravitational anomalies, Commun. Math. Phys. 100 (1985), 197-229. MR 87k:58282
  • 52. -, Supersymmetry and Morse theory, J. Differential Geometry 17 (1982), 661-692. MR 84b:58111

Similar Articles

Retrieve articles in Bulletin of the American Mathematical Society with MSC (1991): 58G25, 58G10, 58G25, 58G26

Retrieve articles in all journals with MSC (1991): 58G25, 58G10, 58G25, 58G26


Additional Information

Steven Rosenberg
Affiliation: Department of Mathematics, Boston University, Boston, Massachusetts 02215
Email: sr@math.bu.edu

DOI: https://doi.org/10.1090/S0273-0979-97-00731-3
Additional Notes: Partially supported by the NSF
Article copyright: © Copyright 1997 American Mathematical Society

American Mathematical Society